Homework 6
Math 352, Fall 2011
Instructions: Solve both of these problems. Your solutions must be written in LATEX.
Due Date: Friday, November 4
1. Let C be the cylinder x2 + y 2 = 1, and let f : C → R3 be the function
f (x, y, z) = (x cos z, y cos z, sin z).
Recall that the vectors t1 = (−y, x, 0) and t2 = (0, 0, 1) are a basis for the tangent
space to C at each point.
(a) Find the 3 × 2 matrix for Df with respect to the basis {t1 , t2 }.
(b) Compute the set of all critical points for f . Describe this set geometrically.
(c) Prove that the image of f is precisely the unit sphere S 2 .
(d) Find the preimages under f of the points (0, 0, 1), (0, 0, −1), and (1, 0, 0). Describe
these preimages geometrically.
2. Use the method of Lagrange multipliers to solve the following problems:
(a) Find the four critical points for the function f (x, y, z) = xy + z 2 on the cylinder
x2 + y 2 = 1.
(b) Find the six critical points for the function g(x, y, z) = xz + yz on the unit sphere.